End Model Behaviors: Disagreement Between Teacher & Student

[KingNothing]

A few days ago, my teacher and I had a disagreement about end model behaviors. There was a question on the test asking for the two end model behaviors of the following function:

http://img219.imageshack.us/my.php?image=graph1be.gif

My answers were 2^x for the REMB and -x*sin(x) for the LEMB.

He claims that the only choices for the LEMB are -x and sin(x). His logic is not mathematically sound, because the sin(x) and -x have an equal effect on the graph. He simply claims it "looks more like sin(x)" becuase f(x)=-x does not have a wave-like look to it. At the same time, he admits that sin(x) does not have a growing amplitude. He says that it can't be -x*sin(x) because they are separated by multiplication.

I also disagree with him because on another EMB problem, the correct answer was 3x. According to his logic on the aforementioned problem, the choices should be "3" or "x", yet on this one he agrees with 3x.

What do you guys think?

 

[arildno]

We are talking asymptotic behaviours of some function here, right?
Post the damn function.

 

[HallsofIvy]

Arildno, the function and its graph are given in the attachment. It is
f(x)= x sin(x)+ e^x.

KingNothing: appeal to the precise definition of "End Model Behavior". I don't see what "separated by multiplication" has to do with anything.

 

[arildno]

When I look at the attachment, there seems to say
$$f(x)=x^{x}\sin(x)2^{x}x$$
which doesn't give any meaning for negative x's.

 

[KingNothing]

Look at the upper-right corner of this image:
http://img219.imageshack.us/img219/4562/graph1be.gif

HallsofIvy, what is the precise definition of end model behavior?

 

[HallsofIvy]

It's your problem! How could you expect to be able to answer any question about "End Model Behavior" if you don't know what it means? What does your textbook say!

 

[KingNothing]

We don't have a textbook. The class is Calc 2, we are in the process of reviewing Calc 1. All I'm asking for is a good link to a site that clearly defines End Model Behavior.

 

[arildno]

I'll assume that you are after the asymptotic behaviours at left/right infinities for the function: $$f(x)=x\sin(x)+2^{x}$$
Short answer: You are right, your teacher shouldn't have been a teacher.
Towards minus infinity, $2^{x}$ becomes subdominant to the term $$x\sin(x)$$ over most of the line apart from tiny, shrinking regions around the zeroes of sin(x) where the exponential will continue to be important. (It will, for example, influence the actual locations of the functions zeroes)

Ignoring that quibble, xsin(x) yields for the most part the asymptotic behaviour of the function as x trundles along to minus infinity.